What is the integral from 1 to 2 of arctan(x) ?

The integral from 1 to 2 of arctan(x) is 2arctan(2)-1/2 ln(5)-pi/4+1/2 ln(2)

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integral from 1 to 2 of arctan(x)

\int_{1}^{2} arctan (x) d x

2 arctan (2) - \frac{1}{2} ln (5) - \frac{π}{4} + \frac{1}{2} ln (2)

Decimal

0.97075 \dots

Solution steps

\int_{1}^{2} arctan (x) d x

Apply Integration By Parts

= [x arctan (x) - \int \frac{x}{x ^{2} + 1} d x]_{1}^{2}

\int \frac{x}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1

= [x arctan (x) - \frac{1}{2} ln x^{2} + 1]_{1}^{2}

Compute the boundaries:

2 arctan (2) - \frac{1}{2} ln (5) - \frac{π}{4} + \frac{1}{2} ln (2)

= 2 arctan (2) - \frac{1}{2} ln (5) - \frac{π}{4} + \frac{1}{2} ln (2)

What is the integral from 1 to 2 of arctan(x) ?
The integral from 1 to 2 of arctan(x) is 2arctan(2)-1/2 ln(5)-pi/4+1/2 ln(2)